Falling liquid films are thin viscous liquid flows driven down an inclined plate by gravity. For several decades, they have been under extensive investigation due to their important role in various mechanical and chemical applications, such as complex flow coating [1]} and cooling of mechanical and electronic systems [2]}. For instance, the wavy nature of falling films organized into localised structures or solitary waves [3]} were found to increase the heat/mass transfer rate by \(10-100\%\)  compared to flat films [4]}. Therefore, understanding the mechanisms leading to the complex dynamics of falling liquid films are crucial to the development of various industrial applications. Studying the linear stability is considered the first step in understanding the disordered spatio-temporal state that characterizes falling liquid films. Investigating the onset of the instability and the domains in which the flow is stable or unstable is crucial to understand this kind of flows.
